Semiconductors, Insulators and Metals

Transcription

1 CHAPTER 2 ENERGY BANDS AND EFFECTIVE MASS Semiconductors, insulators and metals Semiconductors Insulators Metals The concept of effective mass Prof. Dr. Beşire GÖNÜL

2 Semiconductors, Insulators and Metals The electrical properties of metals and insulators are well known to all of us. Everyday experience has already taught us a lot about the electrical properties of metals and insulators. But the same cannot be said about semiconductors. What happens when we connect a battery to a piece of a silicon; would it conduct well? or would it act like an insulator?

3 The name semiconductor implies that it conducts somewhere between the two cases (conductors or insulators) Conductivity : σ σmetals ~10 10 /Ω-cm S/C σinsulators ~ /Ω-cm The conductivity (σ) of a semiconductor (S/C) lies between these two extreme cases.

4 .:: The Band Theory of Solids :: N Forbidden Forbidden Number of atoms Allowed Allowed Allowed The electrons surrounding a nucleus have certain welldefined energy-levels. Electrons don t like to have the same energy in the same potential system. The most we could get together in the same energy-level was two, provided thet they had opposite spins. This is called Pauli Exclusion Principle.

5 The difference in energy between each of these smaller levels is so tiny that it is more reasonable to consider each of these sets of smaller energy-levels as being continuous s of energy, rather than considering the enormous number of discrete individual levels. Each allowed is seperated from another one by a forbidden. Electrons can be found in allowed s but they can not be found in forbidden s.

6 Consider 1 cm 3 of Silicon. How many atoms does this contain?.:: CALCULATION Solution: The atomic mass of silicon is 28.1 g which contains Avagadro s number of atoms. Avagadro s number N is 6.02 x atoms/mol. The density of silicon: 2.3 x 10 3 kg/m 3 so 1 cm 3 of silicon weighs 2.3 gram and so contains = atoms This means that in a piece of silicon just one cubic centimeter in volume, each electron energy-level has split up into 4.93 x smaller levels!

7 .:: Semiconductor, Insulators, Conductors ::. Full Empty All energy levels are occupied by electrons All energy levels are empty ( no electrons) Both full and empty s do not partake in electrical conduction.

8 .:: Semiconductor energy s at low temperature ::. At low temperatures the valance is full, and the conduction is empty. Electron en nergy Forbidden energy gap [Eg] Empty conduction Full valance Recall that a full can not conduct, and neither can an empty. At low temperatures, s/c s do not conduct, they behave like insulators. The thermal energy of the electrons sitting at the top of the full is much lower than that of the Eg at low temperatures.

9 Conduction Electron : Assume some kind of energy is provided to the electron (valence electron) sitting at the top of the valance. This electron gains energy from the applied field and it would like to move into higher energy states. This electron contributes to the conductivity and this electron is called as a conduction electron. Forbidden energy gap [Eg] Empty conduction Full valance At 0 0 K, electron sits at the lowest energy levels. The valance is the highest filled at zero kelvin.

10 Semiconductor energy s at room temperature When enough energy is supplied to the e - sitting at the top of the valance, e - can make a transition to the bottom of the conduction. When electron makes such a transition it leaves behind a missing electron state. This missing electron state is called as a hole. Hole behaves as a positive charge carrier. Magnitude of its charge is the same with that of the electron but with an opposite sign. energy +e - +e - +e - +e - Empty conduction Forbidden energy gap [Eg] Full valance

11 Conclusions ::. Holes contribute to current in valance (VB) as e - s are able to create current in conduction (CB). Hole is not a free particle. It can only exist within the crystal. A hole is simply a vacant electron state. A transition results an equal number of e - in CB and holes in VB. This is an important property of intrinsic, or undoped s/c s. For extrinsic, or doped, semiconductors this is no longer true.

12 Bipolar (two carrier) conduction Electron energy empty occupied After transition Valance Band (partly filled ) After transition, the valance is now no longer full, it is partly filled and may conduct electric current. The conductivity is due to both electrons and holes, and this device is called a bipolar conductor or bipolar device.

13 What kind of excitation mechanism can cause an e - to make a transition from the top of the valance (VB) to the minimum or bottom of the conduction (CB)? Answer : Thermal energy? Electrical field? Electromagnetic radiation? Eg Partly filled CB Partly filled VB Energy diagram of a s/c at a finite temperature. To have a partly field configuration in a s/c, one must use one of these excitation mechanisms.

14 1-Thermal Energy : Thermal energy = k x T = 1.38 x J/K x 300 K =25 mev Excitation rate = constant x exp(-eg / kt) Although the thermal energy at room temperature, RT, is very small, i.e. 25 mev, a few electrons can be promoted to the CB. Electrons can be promoted to the CB by means of thermal energy. This is due to the exponential increase of excitation rate with increasing temperature. Excitation rate is a strong function of temperature.

15 2- Electric field : For low fields, this mechanism doesn t promote electrons to the CB in common s/c s such as Si and GaAs. An electric field of V/m can provide an energy of the order of 1 ev. This field is enormous. So, the use of the electric field as an excitation mechanism is not useful way to promote electrons in s/c s.

16 3- Electromagnetic Radiation : c E= hν = h = (6.62 x10 J s) x(3 x10 m/ s)/ λ( m) E( ev) = λ λ (in µ m) h = 6.62 x J-s c = 3 x 10 8 m/s 1 ev=1.6x10-19 J Near infrared 1.24 for Silicon Eg = 1.1 ev λ ( µ m) = = 1.1µ m 1.1 To promote electrons from VB to CB Silicon, the wavelength of the photons must 1.1 µm or less

17 Conduction Band e - + Valance Band photon The converse transition can also happen. An electron in CB recombines with a hole in VB and generate a photon. The energy of the photon will be in the order of Eg. If this happens in a direct -gap s/c, it forms the basis of LED s and LASERS.

18 Insulators : The magnitude of the gap determines the differences between insulators, s/c s and metals. The excitation mechanism of thermal is not a useful way to promote an electron to CB even the melting temperature is reached in an insulator. Even very high electric fields is also unable to promote electrons across the gap in an insulator. CB (completely empty) Eg~several electron volts VB (completely full) Wide gaps between VB and CB

19 Metals : Touching VB and CB CB VB CB VB Overlapping VB and CB These two s looks like as if partly filled s and it is known that partly filled s conducts well. This is the reason why metals have high conductivity. No gap between valance and conduction

20 The Concept of Effective Mass : Comparing Free e - in vacuum In an electric field m o =9.1 x Free electron mass An e - in a crystal In an electric field In a crystal m =? If the same magnitude of electric field is applied to both electrons in vacuum and inside the crystal, the electrons will accelerate at a different rate from each other due to the existence of different potentials inside the crystal. The electron inside the crystal has to try to make its own way. So the electrons inside the crystal will have a different mass than that of the electron in vacuum. m * effective mass This altered mass is called as an effective-mass.

21 What is the expression for m * Particles of electrons and holes behave as a wave under certain conditions. So one has to consider the de Broglie wavelength to link partical behaviour with wave behaviour. Partical such as electrons and waves can be diffracted from the crystal just as X-rays. Certain electron momentum is not allowed by the crystal lattice. This is the origin of the energy gaps. n λ=2d sinθ n = the order of the diffraction λ = the wavelength of the X-ray d = the distance between planes θ = the incident angle of the X-ray beam

22 nλ = 2d The waves are standing waves λ = 2π k The momentum is P = hk (1) is the propogation constant The energy of the free electron can be related to its momentum E E P (2) 2 h P = = λ 2m free e - mass, m = h = h 2m λ 2m h h= 2π E = 2m h 2k 2 k (2 π ) The energy of the free e - is related to the k By means of equations (1) and (2) certain e - momenta are not allowed by the crystal. The velocity of the electron at these momentum values is zero. Energy momentum k E versus k diagram is a parabola. Energy is continuous with k, i,e, all energy (momentum) values are allowed. E versus k diagram or Energy versus momentum diagrams

23 To find effective mass, m * We will take the derivative of energy with respect to k ; 2 d E k = h d k m d d k E h m = - m* is determined by the curvature of the E-k curve - m* is inversely proportional to the curvature Change m* instead of m m * = h d E d k This formula is the effective mass of an electron inside the crystal.

24 Direct an indirect- gap materials : Direct- gap s/c s (e.g. GaAs, InP, AlGaAs) CB E For a direct- gap material, the minimum of the conduction and maximum of the valance lies at the same momentum, k, values. e - + k When an electron sitting at the bottom of the CB recombines with a hole sitting at the top of the VB, there will be no change in momentum values. VB Energy is conserved by means of emitting a photon, such transitions are called as radiative transitions.

25 Indirect- gap s/c s (e.g. Si and Ge) VB E + e - k Eg CB For an indirect- gap material; the minimum of the CB and maximum of the VB lie at different k-values. When an e - and hole recombine in an indirect- gap s/c, phonons must be involved to conserve momentum. Phonon Atoms vibrate about their mean position at a finite temperature.these vibrations produce vibrational waves inside the crystal. Phonons are the quanta of these vibrational waves. Phonons travel with a velocity of sound. Their wavelength is determined by the crystal lattice constant. Phonons can only exist inside the crystal.

26 The transition that involves phonons without producing photons are called nonradiative (radiationless) transitions. These transitions are observed in an indirect gap result in inefficient photon producing. s/c and So in order to have efficient LED s and LASER s, one should choose materials having direct gaps such as compound s/c s of GaAs, AlGaAs, etc

27 .:: CALCULATION For GaAs, calculate a typical ( gap) photon energy and momentum, and compare this with a typical phonon energy and momentum that might be expected with this material. photon phonon E(photon) = Eg(GaAs) = 1.43 ev υ E(photon) = h = hc / λ c= 3x10 8 m/sec E(phonon) = h υ= hv s / λ = hv s / a0 λ (phonon) ~a0 = lattice constant =5.65x10-10 m P = h / λ h=6.63x10-34 J-sec Vs= 5x10 3 m/sec ( velocity of sound) λ (photon)= 1.24 / 1.43 = 0.88 µm P(photon) = h / λ = 7.53 x kg-m/sec E(phonon) = hv s / a 0 =0.037 ev P(phonon)= h / λ = h / a 0 = 1.17x10-24 kg-m/sec

28 Photon energy = 1.43 ev Phonon energy = 37 mev Photon momentum = 7.53 x kg-m/sec Phonon momentum = 1.17 x kg-m/sec Photons carry large energies but negligible amount of momentum. On the other hand, phonons carry very little energy but significant amount of momentum.

29 Positive and negative effective mass Direct- gap s/c s (e.g. GaAs, InP, AlGaAs) CB e - + h 2 m E * = 2 2 k d E dk The sign of the effective mass is determined directly from the sign of the curvature of the E-k curve. The curvature of a graph at a minimum point is a positive quantity and the curvature of a graph at a maximum point is a negative quantity. Particles(electrons) sitting near the minimum have a positive effective mass. VB Particles(holes) sitting near the valence maximum have a negative effective mass. A negative effective mass implies that a particle will go the wrong way when an extrernal force is applied.

30 4 GaAs Conduction 4 Si Conduction En nergy (ev) 1 0 E=0.31 Eg En nergy (ev) 1 0 Eg Valance [111] 0 [100] k -2 Valance [111] 0 [100] k Energy structures of GaAs and Si

31 4 3 GaAs Conduction Band gap is the smallest energy separation between the valence and conduction edges. 2 En nergy (ev) 1 0 E=0.31 Eg The smallest energy difference occurs at the same momentum value -1-2 Valance [111] 0 [100] k Direct gap semiconductor Energy structure of GaAs

32 The smallest energy gap is between the top of the VB at k=0 and one of the CB minima away from k= Si Conduction Indirect gap semiconductor Band structure of AlGaAs? En nergy (ev) 1 0 Eg Effective masses of CB satellites? Heavy- and light-hole masses in VB? -1-2 Valance [111] 0 [100] k Energy structure of Si

33 E E g direct transition k

34 E E g direct transition k

35 E E g k

36 E E g indirect transition k

37 E E g indirect transition k